Electronic Bell Construction and Circuit Analysis

Sean E. O'Connor

Introduction

This circuit is based on the article
Percussion Instrument Synthesizer by Forrest M. Mims, Popular Electronics, March, 1976, pgs. 90 - 102.
which describes an audio amplifier with a twin-T notch filter in the feedback loop,
The R-C network gives a sharp peak at a single audio frequency at which feedback occurs.
Disturbing the touch plate introduces a transient which starts a damped oscillation "ringing".
I've added a Hall effect sensor on the input instead of a touch plate to inject the transient, and an audio power amplifer on the output.

Analyzing the RC Network

Let's analyze the network in isolation and add the amplifier and input later.

Kirchhoff's Laws

First apply Kirchhoff's current law, which is equivalent to convervation of charge, on nodes 1-4 gives

$i_1 - i_3 - i_6 = 0$

$i_4 - i_2 + i_7 = 0$

$i_3 - i_4 - i_5 = 0$

$i_6 - i_7 - i_8 = 0$

Recall the convention that currents flowing into the node are taken as positive.

First an aside, recall the Laplace transform of a function $f$ is defined to be
$\mathscr{L}\{ f(t) \} = \int_0^{\infty} f(t) e^{-st} dt$
and has the property
$\mathscr{L}\{ f'(t) \} = s \mathscr{L} \{ f(t) \} - f(0)$
and that the transform is linear.

Now we apply Kirchhoff's voltage law, which is equivalent to convervation of energy (think of following the work W = q dV done
by a charge going around a closed loop), on the meshes 1-4.
But first we apply the Laplace transform to all voltage terms in the loop using linearity and assuming initial currents and
voltages are zero.
For simplicity of notation, let's use $i(t)$ to denote $\mathscr{L} \{ i(t) \}$, and $V(t)$ to denote $\mathscr{L} \{ V(t) \}$.

Ringing the Bell

To ring the bell in operation, one applies a sharp impulse to the input.
To approximate this, set $e(t) = \delta(t)$, the Dirac delta function.
Then $e(s) = \mathscr{L}\{ \delta(t) \} = 1$
The solution will be $V_2(t) = \mathscr{L}^{-1}\{ H(s) \}$